## CIRCULAR MOTION

• CIRCULAR MOTION
• When a particle moves in a plane such that its distance from a fixed (or moving) point
remains constant, then its motion is known as circular motion with respect to that fixed
(or moving) point

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• KINEMATICS OF CIRCULAR MOTION
• To decide the angular position of a point in space we need to specify (i) origin and (ii) reference line.
The angle made by the position vector w.r.t. origin, with the reference line is called angular position.
Clearly angular position depends on the choice of the origin as well as the reference line.

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• Important points
• Infinitesimally small angular displacement is a vector quantity, but finite angular displacement is a
scalar, because while the addition of the Infinitesimally small angular displacements is commutative,
addition of finite angular displacement is not.

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• Angular Velocity
• Instantaneous Angular Velocity
It is the limit of average angular velocity as ?t approaches zero. i.e.

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• Important points
• Angular velocity has dimension of [T-1] and SI unit rad/s.

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• Angular Acceleration
• Instantaneous Angular Acceleration :
It is the limit of average angular acceleration as ?t approaches zero, i.e.,

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• Motion with constant angular velocity
• ? ? Angular displacement
Circular motion with constant angular acceleration is analogous to one dimensional translational motion with
constant acceleration. Hence even here equation of motion have same form.

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• RELATION BETWEEN SPEED AND ANGULAR VELOCITY
• RELATION BETWEEN SPEED AND ANGULAR VELOCITY

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• RELATIVE ANGULAR VELOCITY
• Just as velocities are always relative, similarly angular velocity is also always relative. There is no such thing as
absolute angular velocity. Angular velocity is defined with respect to origin, the point from which the position
vector of the moving particle is drawn.

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• RADIAL AND TANGENTIAL ACCELERATION
• There are two types of acceleration in circular motion ; Tangential acceleration and centripetal acceleration

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• IMPORTANT POINT
• Differentiation of speed gives tangential acceleration

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• Calculation of centripetal acceleration
• Consider a particle which moves in a circle with constant speed v as shown in figure

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• DYNAMICS OF CIRCULAR MOTION
• If there is no force acting on a body it will move in a straight line (with constant speed). Hence if a body is moving
in a circular path or any curved path, there must be some force acting on the body.
If speed of body is constant, the net force acting on the body is along the inside normal to the path of the body
and it is called centripetal force.

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• RADIUS OF CURVATURE
• Any curved path can be assumed to be made of infinite circular arcs. Radius of curvature at a point is the radius
of the circular arc at a particular point which fits the curve at that point.

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• MOTION IN A VERTICAL CIRCLE
• Let us consider the motion of a point mass tied to a string of
length ï¬ and whirled in a vertical circle. If at any time the body
is at angular position ?, as shown in the figure, the forces
acting on it are tension T in the string along the radius towards
the center and the weight of the body mg acting vertically
down wards.

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• CONDITION FOR OSCILLATION OR LEAVING THE CIRCLE
• vertical plane. In this case, the motion of the point mass which depend on
â€˜whether tension becomes zero before speed becomes zero or vice versa

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• CONDITION FOR LOOPING THE LOOP IN SOME OTHER CASES
• CONDITION FOR LOOPING THE LOOP IN SOME OTHER CASES
Case 1 : A mass moving on a smooth vertical circular track.

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• CIRCULAR TURNING ON ROADS
• When vehicles go through turnings, they travel along a nearly circular
arc. There must be some force which will produce the required centripetal
acceleration. If the vehicles travel in a horizontal circular path,
this resultant force is also horizontal. The necessary centripetal force
is being provided to the vehicles by following three ways.

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• By Friction Only
• Suppose a car of mass m is moving at a speed v in a horizontal circular arc of radius r. In this case,
the necessary centripetal force to the car will be provided by force of friction f acting towards center

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• By Banking of Roads Only
• Friction is not always reliable at circular turns if high speeds and sharp turns are involved to avoid dependence on
friction, the roads are banked at the turn so that the outer part of the road is some what lifted compared to the
inner part.

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• By Friction and Banking of Road Both
• If a vehicle is moving on a circular road which is rough and banked also, then three forces may act on the vehicle,
of these the first force, i.e., weight (mg) is fixed both in magnitude and direction.

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• Note
• The expression tan ? = rg
v2
also gives the angle of banking for an aircraft, i.e., the angle through
which it should tilt while negotiating a curve, to avoid deviation from the circular path.

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• CENTRIFUGAL FORCE
• When a body is rotating in a circular path and the centripetal force vanishes, the body would leave the circular
path. To an observer A who is not sharing the motion along the circular path, the body appears to fly off tangentially
at the point of release. To another observer B, who is sharing the motion along the circular path (i.e., the
observer B is also rotating with the body which is released, it appears to B, as if it has been thrown off along the
radius away from the centre by some force. This inertial force is called centrifugal force.)

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• EFFECT OF EARTHS ROTATION ON APPARENT WEIGHT
• The earth rotates about its axis at an angular speed of one revolution per 24
hours. The line joining the north and the south poles is the axis of rotation.
Every point on the earth moves in a circle. A point at equator moves in a circle of
radius equal to the radius of the earth and the centre of the circle is same as the
centre of the earth. For any other point on the earth, the circle of rotation is
smaller than this. Consider a place P on the earth (figure).

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